Proposition 12

Let E, F be two vector spaces over a field K and $u : E \to F$ a linear mapping. For the linear equation

to have at least one solution $x \in E$, it is necessary and sufficient that $y_0$ be orthogonal to the kernel of the transpose mapping ${}^t u$.

The orthogonal of $u(E)$ in $F^*$ is ${}^t u^{-1}(0)$ (§ 2, no. 5, Corollary to Proposition 8) and the orthogonal of ${}^t u^{-1}(0)$ in F is therefore $u(E)$ (no. 5, Theorem 7 (ii)).

We shall obtain a more convenient criterion for systems of scalar linear equations

where the unknown $x$ takes its values in a vector space E over a field K, the $x_i^*$ are linear forms on E and the right hand sides $\eta_i$ elements of K.

If we consider a basis $(a_\lambda)_{\lambda \in L}$ of E, the system (21) is equivalent to the system of equations

with $x = \sum_{\lambda \in L} \xi_\lambda a_\lambda$, the solutions of (22) being of necessity families $(\xi_\lambda)$ of elements of K of finite support.